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## Lattice QCD for Exotic Hadrons

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**Lattice QCD for Exotic Hadrons**H. Suganuma (Kyoto Univ.) K. Tsumura (Kyoto U./ Fuji Film) N. Ishii (Tsukuba Univ.) T.T. Takahashi (Tokyo I.Tech.) F. Okiharu (Nihon Univ.) A. Yamamoto (Kyoto Univ.) H. Iida (RCNP, Osaka U.) Challenge to New Exotic Hadrons with Heavy QuarksJuly 7-8 2007, RCNP Osaka**Introduction**・Exotic hadrons (multi-quarks, hybrids, glueballs) have been interesting objects beyond the simple quark model. ・After the experimental report on Θ+(1530) at SPring-8, lots of theoretical analyses for the exotic hadrons have been done or revised. ・Recently, Tetra-Quark has been investigated as an interesting subject in QCD and hadron physics, because several charmed tetra-quark candidates have been experimentally discovered in these years. ・This subject, Tetra-Quark, also relates to the old famous “scalar meson puzzle” in the light-quark sector.**Exotic Hadrons：New-type of quantum many-body system**near D-D* threshold Experimental discovery of Exotic Charmed Hadron candidates have been done at KEK（Belle）, SLAC（BaBar）: Tetra-Quark or Hybrid candidates X(3872), Y(3940), Ds0+(2317) etc. → This finding gives a Strong Impact to QCD and quark-hadron physics Figs. from KEK web-site unusual decay pattern**From Prof. A. Bonder’s Talks**So many new-type hadrons which cannot be explained with the simple quark model have been observed. → We need theoretical Analysis of Multi-Quarks or Hybrids based on QCD**“Scalar Meson Puzzle” and Tetra-quark Candidates in the**Light-Quark Sector Also in the light-quark sector, there are tetra-quark candidates. ・There are five 0++ isoscalar mesonsbelow 2GeV: f0(400-1200), f0(980), f0(1370), f0(1500) and f0(1710). ・Among them, f0(1500) and f0(1710) are expected to be the lowest scalar glueball or an ss scalar meson. ・f0(1370) is considered as the lowest qq scalar meson in the quark model. For, in the quark model, the lowest qq scalar meson is 3P0, and therefore it turns to be rather heavy. ・So, what are the two light scalar mesons, f0(400-1200) and f0(980)? This is the “scalar meson puzzle”, which is unsolved even at present. ・As a possible answer, Jaffe proposed tetraquark (qqqq)assignment for low-lying scalar mesons such as f0(980) and a0(980)in 1977. ・Then, we examine the scalar tetraquark f0(udud) in the light-quark sector in O(a)-improvedanisotropic lattice QCD (high-resolution in time direction). Here, we only consider connected diagrams at the quenched level. － － － － － － －**“Scalar Meson Puzzle” and Tetra-quark Candidates in the**Light-Quark Sector Also in the light-quark sector, there are tetra-quark candidates. ・There are five 0++ isoscalar mesonsbelow 2GeV: f0(400-1200), f0(980), f0(1370), f0(1500) and f0(1710). ・Among them, f0(1500) and f0(1710) are expected to be the lowest scalar glueball or an ss scalar meson. ・f0(1370) is considered as the lowest qq scalar meson in the quark model. For, in the quark model, the lowest qq scalar meson is 3P0, and therefore it turns to be rather heavy. ・So, what are the two light scalar mesons, f0(400-1200) and f0(980)? This is the “scalar meson puzzle”, which is unsolved even at present. ・As a possible answer, Jaffe proposed tetraquark (qqqq)assignment for low-lying scalar mesons such as f0(980) and a0(980)in 1977. ・Then, we examine the scalar tetraquark f0(udud) in the light-quark sector in O(a)-improvedanisotropic lattice QCD (high-resolution in time direction). Here, we only consider connected diagrams at the quenched level. － － － － － － －**“Scalar Meson Puzzle” and Tetra-quark Candidates in the**Light-Quark Sector Also in the light-quark sector, there are tetra-quark candidates. ・There are five 0++ isoscalar mesonsbelow 2GeV: f0(400-1200), f0(980), f0(1370), f0(1500) and f0(1710). ・Among them, f0(1500) and f0(1710) are expected to be the lowest scalar glueball or an ss scalar meson. ・f0(1370) is considered as the lowest qq scalar meson in the quark model. For, in the quark model, the lowest qq scalar meson is 3P0, and therefore it turns to be rather heavy. ・So, what are the two light scalar mesons, f0(400-1200) and f0(980)? This is the “scalar meson puzzle”, which is unsolved even at present. ・As a possible answer, Jaffe proposed tetraquark (qqqq)assignment for low-lying scalar mesons such as f0(980) and a0(980)in 1977. ・Then, we examine the scalar tetraquark f0(udud) in the light-quark sector in O(a)-improvedanisotropic lattice QCD (high-resolution in time direction). Here, we only consider connected diagrams at the quenched level. － － － － － － －**Theoretical Conjecture for Light Tetra-Quark Systems**Pseudo-scalar meson Scalar meson Simple quark model JP=0- JP=0+ － － q q q q － q and q has opposite parity ↓ Parity of q q meson : P=(-1)L+1 － L=0 L=1 light should be heavy － qq scalar meson is P-wave (L=1) , and hence its mass should be large. ~ Particle Data Group identifies qq scalar meson as f0(1370), a0(1450) . －**－**qq scalar meson in Lattice QCD H. Suganuma et al. (2005) 1.3~1.4GeV － qq scalar meson mass is about1.3~1.4GeV→f0(1370), a0(1450)**“Scalar Meson Puzzle” and Tetra-quark Candidates in the**Light-Quark Sector Also in the light-quark sector, there are tetra-quark candidates. ・There are five 0++ isoscalar mesonsbelow 2GeV: f0(400-1200), f0(980), f0(1370), f0(1500) and f0(1710). ・Among them, f0(1500) and f0(1710) are expected to be the lowest scalar glueball or an ss scalar meson. ・f0(1370) is considered as the lowest qq scalar meson in the quark model. For, in the quark model, the lowest qq scalar meson is 3P0, and therefore it turns to be rather heavy. ・So, what are the two light scalar mesons, f0(400-1200) and f0(980)? This is the “scalar meson puzzle”, which is unsolved even at present. ・As a possible answer, Jaffe proposed tetraquark (qqqq)assignment for low-lying scalar mesons such as f0(980) and a0(980)in 1977. ・Then, we examine the scalar tetraquark f0(udud) in the light-quark sector in O(a)-improvedanisotropic lattice QCD (high-resolution in time direction). Here, we only consider connected diagrams at the quenched level. － － － － － － －**“Scalar Meson Puzzle” and Tetra-quark Candidates in the**Light-Quark Sector Also in the light-quark sector, there are tetra-quark candidates. ・There are five 0++ isoscalar mesonsbelow 2GeV: f0(400-1200), f0(980), f0(1370), f0(1500) and f0(1710). ・Among them, f0(1500) and f0(1710) are expected to be the lowest scalar glueball or an ss scalar meson. ・f0(1370) is considered as the lowest qq scalar meson in the quark model. For, in the quark model, the lowest qq scalar meson is 3P0, and therefore it turns to be rather heavy. ・So, what are the two light scalar mesons, f0(400-1200) and f0(980)? This is the “scalar meson puzzle”, which is unsolved even at present. ・As a possible answer, Jaffe proposed tetraquark (qqqq)assignment for low-lying scalar mesons such as f0(980) and a0(980)in 1977. ・Then, we examine the scalar tetraquark f0(udud) in the light-quark sector in O(a)-improvedanisotropic lattice QCD (high-resolution in time direction). Here, we only consider connected diagrams at the quenched level. － － － － － － －**Theoretical Conjecture for Light Tetra-Quark Systems**Pseudo-scalar meson Scalar meson Simple quark model JP=0- JP=0+ － － q q q q － q and q has opposite parity ↓ Parity of q q meson : P=(-1)L+1 － L=0 L=1 light should be heavy － qq scalar meson is P-wave (L=1) , and hence its mass should be large. ~ Particle Data Group identifies qq scalar meson as f0(1370), a0(1450) . － ・f0(980), a0(980), σ(600)：light scalar mesons ~ tetra-quark candidates anti-quark quark π σ π or ～ anti-quark quark hadronic molecule tetra-quark state**“Scalar Meson Puzzle” and Tetra-quark Candidates in the**Light-Quark Sector Also in the light-quark sector, there are tetra-quark candidates. ・There are five 0++ isoscalar mesonsbelow 2GeV: f0(400-1200), f0(980), f0(1370), f0(1500) and f0(1710). ・Among them, f0(1500) and f0(1710) are expected to be the lowest scalar glueball or an ss scalar meson. ・f0(1370) is considered as the lowest qq scalar meson in the quark model. For, in the quark model, the lowest qq scalar meson is 3P0, and therefore it turns to be rather heavy. ・So, what are the two light scalar mesons, f0(400-1200) and f0(980)? This is the “scalar meson puzzle”, which is unsolved even at present. ・As a possible answer, Jaffe proposed tetraquark (qqqq)assignment for low-lying scalar mesons such as f0(980) and a0(980)in 1977. ・Then, we examine the scalar tetraquark f0(udud) in the light-quark sector in O(a)-improvedanisotropic lattice QCD (high-resolution in time direction). Here, we only consider connected diagrams at the quenched level. － － － － － － －**Calculation of Light Scalar 4Q state in Anisotropic Quenched**Lattice QCD Note that, in Quenched Lattice QCD, the 4Q state component can be investigated without suffering from the mixture of qq component, because there is no dynamical quark annihilation and creation in quenched QCD. ~ a sort of “merit” of quenched approximation for the study of multi-quark hadrons Each nQ component can be investigated individually to some extent in quenched QCD. - In Ｆｕｌｌ Lattice QCD, the 4Q state component is inevitably mixed with theqq component due to the appearance of dynamical quark loops. -**Gauge part**O(a)-improved Anisotropic Lattice QCD Quark part : clover fermion ~ O(a)-improved i.e. discretized error is reduced (a: lattice spacing) β=5.75 on 123× 96 with renormalized anisotropy as/at=4 This leads to the spatial/temporal lattice spacing as as = 0.18fm (as-1 = 1.10GeV), at = 0.045fm (at-1= 4.40GeV). The spatial lattice size is L =12 as = 2.15 fm. (Sommer scale: r0-1=395MeV) happing parameter: κ=0.1210 ~ 0.1240 (ms~2ms,) → mπ ≧ 650MeV. 1,827 gauge configurations, picked up every 500 sweeps after thermalization of 10,000 sweeps.**Anisotropic Lattice QCD**The mass and the spectral function of the 4Q state are extracted from thetemporal behavior of the 4Q correlator with the 4Q operator To get detailed information on the temporal behavior of the 4Q correlatorD4Q(τ), we adopt anisotropic lattice QCD with high-resolution in the temporal direction.**4Q correlator and effective mass analysis**From the temporal 4Q correlator D4Q(t), the low-lying mass of the 4Q state is estimated by the “effective mass” meff(t) = ln {D4Q(t)/D4Q(t+1)} in large t region. ・Zero-momentum projection is done to remove the total kinetic energy. ・We use the same quark mass for u and d. ・We use spatially-extended operators to enhance the low-lying spectra. ・For temporal direction, we impose Dirichlet boundary condition on the quark fields to avoid the contamination of backward propagations. ・For spatial directions, we use not only standard periodic boundary condition (PBC) but also the hybrid boundary condition (HBC), where qq meson state inevitably has a finite momentum while 4Q state can have zero momentum. -**Hybrid Boundary Condition (HBC)~New method to distinguish**resonance and scattering state In the Hybrid Boundary Condition (HBC) method, anti-periodic and periodic boundary conditions are imposed on quarks and anti-quarks, respectively. By applying the HBC on a finite-volume lattice, the mass of qq meson state is raised up, while the mass of spatially compact 4Q resonance is almost unchanged. - cf.Through the HBC test, low-lying 5Q is clarified to be a NK scattering state. No low-lying Penta-Quark resonances (1/2+, 1/2-, 3/2+, 3/2-) N.Ishii et al. Proc. of Lattice 2004 before null-result experiments reported Physical Review D71 (2005) 034001,Physical Review D71 (2005) 074503**κ=0.1240**Effective Mass Plot for 4Q state threshold HBC threshold PBC**Periodic boundary condition (PBC) case**Effective mass of 4Q state almost coincides with the ππ threshold in PBC case**Hybrid boundary condition (HBC) case**? Effective mass of 4Q state differs from the ππthreshold in HBC case: maximal difference is about 100MeV**MEM analysis to obtain the spectral function from the**temporal correlator To clarify whether the resonance state exits or not, we only have to obtain the spectral function A(ω), and investigate its peak structure. So, we perform MEM analysis to obtain the spectral function A(ω) from the temporal correlator D(τ). MEM analysis reveals the possible surviving of J/ψ even above the critical temperature Tc in lattice QCD. [Asakawa-Hatsuda, Datta et al., Umeda et al., H.Iida, T.Doi, N.Ishii, H.Suganuma and K.Tsumura, Physical Review D74 (2006) 074502 (12 pages).]**Maximum Entropy Method (MEM)**MEM is a useful method to obtain the spectral function A(ω) in a model-independent way from the correlator D(τ) by solving the inverse problem, where the kernel at zero temperature is given by K(τ,ω) ≡ e-τω. Generally, the inverse problem is mathematically difficult, and it is actually difficultto solve the linear simultaneous equation directly with respect to A(ω) both analytically and numerically. In the MEM framework, for the given lattice data of the correlator D(τ),the most probable spectral function A(ω) is extracted, by maximizing the conditional probability P[A|D] of A(ω) for given D(τ), instead of solving the linear simultaneous equation directly with respect to A(ω).**Using the Bayes theorem in probability theory, P[A|D] =**P[D|A] P[A] / P[D], the conditional probability P[A|D] is obtained as a functional with respect to A(ω),P[A|D] ∝eαS[A]-L[A] ・Here, L[A] is the likelihood function, whose minimization gives the ordinary χ2 fit. ( D[A](τ) ≡∫dωK(τ,ω) A(ω), C-1: inverse covariant matrix ) ・S[A] is the Shannon-Jaynes entropy defined as where m(ω) is the default function to give a plausible form of A[ω],e.g., P-QCD form in the high-energy region. In MEM, L[A] tends to be minimized and S[A] to be maximized.**Default Function of 4Q state**Default function m(ω) appearing in the Shannon-Jaynes entropyS[A] gives a plausible form of the spectral function A[ω], e.g., P-QCD form in the high-energy region, which plays a role of a suitable “boundary condition” of A[ω] at large ω. For the local 4Q operator, we calculate the default function m(ω) of the 4Q state at the lowest perturbation of QCD as and we get the final form as**MEM analysis for the 4Q state**with the local 4Q operator From the lattice QCD data for the 4Q correlator we get the spectral function A(ω) for 4Q statethrough the MEM analysis. Note that the appearance of the peak structure in the spectral function A(ω) in the MEM analysis is highly nontrivial, because the default functionm(ω) obtained with P-QCD does not have any peak structure. ~ simple monotonic function With this default function, the peak structure of the spectral function A(ω) is biased to be disappeared in the MEM analysis. → The peak structure surviving in the MEM analysis is mathematically reliable.**Spectral function for 4Q state (PBC) obtained with MEM**ππ threshold**Lattice data for 4Q correlator (PBC) and MEM result**The lattice data for 4Q correlator D(τ) is well reproduced with the spectral function A(ω) obtained by MEM**Spectral function for 4Q state (PBC) obtained with MEM**ππ threshold Sharp peak appears just above ππ threshold (~1.3GeV)→ resonance state ? But, in finite-size lattice, the momentum of the pion is discretized, and all the scattering states are observed as resonance-like states.**Spectral function for 4Q state (HBC) obtained with MEM**ππ threshold Sharp peak appears just below HBCππthreshold (~1.7GeV)The peak position is shifted → resonance state sensitive to BC~ spatially-extended two-pion scattering state**Lattice data for 4Q correlator (HBC) and MEM result**The lattice data for 4Q correlator D(τ) is well reproduced with the spectral function A(ω) obtained by MEM**Spectral function for 4Q state obtained with MEM**Hybrid BC Ordinary Periodic BC The peak position is shifted → resonance state sensitive to BC~ spatially-extended two-pion scattering stateThere is no extra low-lying peak in PBC→ No low-lying four-quark resonance state**Systematical Studies for Multi-Quark Potential in Lattice**QCD “Detailed Analysis of Tetraquark Potential and Flip Flop in SU(3) Lattice QCD”F. Okiharu, H. Suganuma and T.T. Takahashi Physical Review D72 (2005) 014505 (17 pages). “First Study for the Pentaquark Potential in SU(3) Lattice QCD”F. Okiharu, H. Suganuma and T.T. Takahashi Physical Review Letters 94 (2005) 192001 (4 pages). “Detailed Analysis of the Gluonic Excitation in the 3Q System in Lattice QCD”T.T. Takahashi and H. Suganuma Physical ReviewD70 (2004) 074506 (13 pages). “Gluonic Excitation of the Three-Quark System in SU(3) Lattice QCD”T.T. Takahashi and H. Suganuma Physical Review Letters 90 (2003) 182001 (4 pages). “Detailed Analysis of the Three Quark Potential in SU(3) Lattice QCD”T.T. Takahashi, H. Suganuma et al. Physical ReviewD65 (2002) 114509 (19 pages). “Three-Quark Potential in SU(3) Lattice QCD”T.T. Takahashi, H. Suganuma et al. Physical Review Letters 86 (2001) 18-21.**Quark-antiquark static potential in Lattice QCD**M.Creutz (1979,80) G.S.Bali (2001) T.T.Takahashi et al. (2002) JLQCD (2003) V(r) r0=0.5fm:unit quark anti-quark g2 3π 1 r V(r) = － +σr r quark anti-quark － One-dimensional squeezing of color flux between q and q anti-quark quark g g**Physical Review Letters 86 (2001) 18**Physical Review D65 (2002) 114509 Physical Review Letters 90 (2003) Physical Review D70 (2004) 074506 Physical Review D72 (2005) 014505 Baryonic 3 Quark Potential in Lattice QCD quark quark What Shape of Color Flux? Confining Force? quark Before our study, there was almost No lattice QCD study for the Three-Quark Potential**Physical Review Letters 86 (2001) 18**Physical Review D65 (2002) 114509 Physical Review Letters 90 (2003) Physical Review D70 (2004) 074506 Physical Review D72 (2005) 014505 Baryonic 3 Quark Potential in Lattice QCD quark quark What Shape of Color Flux? Confining Force? quark**Physical Review Letters 86 (2001) 18**Physical Review D65 (2002) 114509 Physical Review Letters 90 (2003) Physical Review D70 (2004) 074506 Physical Review D72 (2005) 014505 Baryonic 3 Quark Potential in Lattice QCD conf V3Q(r) quark quark color electric flux quark**Physical Review Letters 86 (2001) 18**Physical Review D65 (2002) 114509 Physical Review Letters 90 (2003) Physical Review D70 (2004) 074506 Physical Review D72 (2005) 014505 Baryonic 3 Quark Potential in Lattice QCD conf V3Q(r) quark quark color electric flux quark Lmin：total length of string linking three valence quarks g2 4π 3 TiaTja |ri - rj| V3Q(r) = ∑ + σLmin i<j One-Gluon-Exchange Coulomb potential Linear potential based on string picture**Lattice QCD result for Color Flux-Tube Formation in baryons**H. Ichie et al., Nucl. Phys. A721, 899 (2003)